Global regularity for systems with p-structure depending on the symmetric gradient

Abstract

In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having p-structure depending only on the symmetric part of the gradient.Keywords. Regularity of weak solutions, symmetric gradient, boundary regularity, natural quantities. MSC. 76A05 (35D35 35Q35)

A Counterexample Related to the Regularity of the p-Stokes Problem

“…Remark 7. Estimates (12) and (14) should clarify the role of the bound ⌊ 3 √ l⌋ from Lemma 1. This bound was chosen with the purpose of making the right-hand side in the final estimate (16) divergent.…”

“…From Remark 8 we know that u possesses the natural regularity (3). However, for this modified problem this regularity property is proved in [29], [12] for any weak solution and any right-hand side f ∈ L p ′ (Ω).…”

“…If these difficulties are separated, one can prove better results. If (1) is considered without the divergence constraint and the resulting pressure gradient, it is proved in [29], [12] that (3) holds for p ∈ (1, 2). If ( 1) is considered with S depending on the full velocity gradient ∇u, it is proved in [17] that u ∈ W 2,r (R 3 ) ∩ W 1,p 0 (R 3 ) for some r > 3, provided p < 2 is very close to 2.…”

Natural second order regularity for systems in the case $1

“…Thus, problem (1) is a generalization to systems of the classical -Laplace problem for scalars Δ := div(|∇ | −2 ∇ ), which corresponds to the case = 0. While the existence of weak solutions is a rather standard result -based on the theory of monotone operators-the regularity of solutions is more complicated and has been addressed for the case 1 < ≤ 2 by Seregin and Shilkin [22] (in the case of a flat boundary) and by the authors of the present paper in [9] (in a general smooth domain). The proof is obtained by a classical strategy: the use of difference quotients to estimate partial derivatives in the tangential directions and ellipticity to recover normal derivatives.…”

The Existence of Strong Solution for Generalized Navier-Stokes Equations withpx-Power Law under Dirichlet Boundary Conditions

“…Global higher differentiability for weak solutions to the problem (1) with pðxÞ = const have been studied by several authors; for example, see [8][9][10][11][12][13][14][15][16][17][18][19] under the condition f ∈ L p∧′ ðΩÞ with p∧ ′ = p/ðp − 1Þ and p ≔ min fp, 2g. It was first established in [8] by Beirao da Veiga.…”